In Leyawiin during Oblivion there is a Dunmer Nine Divine priest by the name of Avrus Adas. He tells you this: " "I used to be a priest of the Tribunal Temple in Kragenmoor. After the collapse, I drifted for a while, until I joined the Chapel. The beggars and the wicked Thieves Guild still trouble me. I wish the Nine Divines offered charity and comfort to the poor like the Temple did."
So the ALMSIVI faithful were obviously demoralised, he even refers to the aftermath of the events of Morrowind as a collapse. AND this was before Vivec disappeared, or the Ministry of Truth came crashing down on Vvardenfell! Vivec's own guess was the ALMSIVI becoming saints instead of being recognised as gods anymore. This I guess probably happened, but going from living gods to a two dead saints and then who knows what happened to Vivec later... well, as I said many were demoralised and as the priest shows at least some left.
I don't think there is any information on what the Dunmer believed during the Oblivion Crisis, or afterwards before Skyrim. Except perhaps in the novels, but I have not read those. My guess is there was a reluctant mix of ALMSIVI saint worship and ancestor cult, more radical Dissident Priests refusing the ALMSIVI, Ashlanders doing what they have always done and finally some Imperial Cult converts.
Certain people were probably disillusioned with religion or felt lost a while, like Avrus Adas.
In Skyrim, as you probably know, you can meet Aphia Velothi on Solstheim. She used to be a priest, but is no longer so but perhaps because of the opposite reason compared to Avrus Adas. You have this little exchange with her:
You're a priest at the Temple? "Far from it. In fact, I'm no longer welcome there." Why? "In all honesty, it came down to a disagreement between what I believe and what Elder Othreloth believes. I'd prefer not to get into the details, but let's just say our opinions about the current status of the Tribunal vary widely."
She doesn't say what the disagreement is, but many speculate she still worshiped the False Tribunal in some way. If she did, the question is whether she considered them gods or saints? I am going out on a limb and guessing she thought of them as saints of the Temple and this opinion became more and more impopular with time. Morrowind's events shattered the belief in the ALMSIVI's divinty, and Vivec's disappearance, Baar Dau falling and time passing diminished their saintly status as well.